For a diffeomorphism \(U\rightarrow U^{\prime }\subset , the quasiconformality condition reduces (locally) to the condition of being contact, i.e. preserving the subbundle \) WT. The study of contact diffeomorphisms has a long history in differential geometry and exterior differential systems going back to Cartan [1]}, [2]}, [3]}, [4]}, [5]}, [6]}.
The particular case considered in this paper – the flag manifold \( equipped with the subbundle \) W\( -- was treated in the papers \cite {cowling_et_al_iwasawa_n_groups,contact_conformal_maps_parabolic_geometry}along with other parabolic geometries.\) Quasiconformal homeomorphisms in \(\mathbb {R}^n\) , \(n\ge 3\) , have been heavily studied since the 1960s. In the Carnot group setting they first appeared in Mostow's rigidity theorem [7]}, and have been investigated intensely since roughly 1990, due to influential contributions of Gromov, Pansu, and applications to geometric group theory. For almost 30 years, these historical threads have merged with other developments in analysis on metric spaces [8]}, [9]} and PDE [10]}, [11]}, [12]}. The rigidity (or regularity) problem for quasiconformal mappings of Carnot groups crystallized over time, first appearing implicitly in the seminal paper [13]},
then in [14]}, [15]}
and finally taking shape with Ottazzi-Warhurst's notion of a rigid Carnot group [6]}, [17]}, explicitly upgraded to a conjecture in [18]}. Corollary REF  settles the Regularity Conjecture for the Carnot group \(N\) . Rigidity of quasiconformal mappings is an instance of rigidity for (solutions to) weak differential inclusions, and more generally partial differential relations, about which there is a vast literature, see for instance [19]}, [12]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}.
